Project 11: Sparse Actuator Placement for Parabolic Control - NMOPT

Goal¶

Design a sparse distributed control for the heat equation. This is an intermediate project that consolidates the nonsmooth material before the more ambitious multiscale project.

Mathematical Formulation¶

Let $Q=\Omega\times(0,T)$ . Solve

\min_{(y,u)} \frac12\int_0^T\|y(t)-y_d(t)\|_{L^2(\Omega)}^2\,dt + \frac\alpha2\int_0^T\|u(t)\|_{L^2(\Omega)}^2\,dt + \beta\int_0^T\|u(t)\|_{L^1(\Omega)}\,dt

(1)

subject to

\begin{cases} y_t-\mu\Delta y+\sigma y=u+f & \text{in }Q,\\ y=0 & \text{on }\partial\Omega\times(0,T),\\ y(0)=y_0 & \text{in }\Omega. \end{cases}

(2)

Optionally impose box constraints

u_a\le u\le u_b.

(3)

The adjoint equation is

\begin{cases} -p_t-\mu\Delta p+\sigma p=y-y_d & \text{in }Q,\\ p=0 & \text{on }\partial\Omega\times(0,T),\\ p(T)=0 & \text{in }\Omega. \end{cases}

(4)

The nonsmooth optimality condition is

0\in \alpha u-p+\beta\partial\|u\|_{L^1} +N_{U_{ad}}(u).

(5)

The proximal-gradient update is cellwise soft-thresholding, followed by box projection if constraints are present.

Implementation Hints¶

Start from ParabolicL1ProximalGradient.
Compare proximal gradient and PDAS.
Use piecewise constant controls so the proximal map is cellwise.
Study how $\beta$ changes the active control region.

Relevant Examples¶

Course code: codes/dealii/include/parabolic_l1_proximal_gradient.h, codes/dealii/include/parabolic_l1_pdas.h.
Course lectures: lecture13.md, lecture16.md.

Deliverables¶

Sparse parabolic control runs for several $\beta$ values.
Plots of support size vs iteration.
Comparison between proximal gradient and PDAS.
Interpretation as actuator placement.